Recently, it has been recognized that the higher derivative dynamics
can be stable at the classical and quantum levels even if the
canonical energy of the model is unbounded. The articles
[1]}, [2]} explain the stability of Pais-Uhlenbeck theory form the
viewpoint of existence of the Hamiltonian form of dynamics with a
bounded (non-canonical) Hamiltonian. The quantization of the model
with an alternative Hamiltonian leads to quantum theory with a
well-defined vacuum state with the lowest energy. The papers
[3]}, [4]} use a special PT-symmetry for construction of
stable quantum mechanics in the higher derivative oscillator model.
The articles [5]}, [6]}, [7]}, [8]} tell us that
the non-linear higher derivative models may have a well-defined
classical dynamics without "collapsing" trajectories with the
runaway behavior. In the last articles, the existence of stable
classical dynamics serves as a necessary prerequisite for
construction of a well-defined quantum theory. For the studies of
field theories, we refer to articles [9]}, [10]}. All the
mentioned in this paragraph models have one common feature: they
admit an alternative (different from the canonical energy) bounded
from below conserved quantities. The quantum stability is achieved
if the model admits Hamiltonian form of dynamics, with the bounded
conserved quantity being the Hamiltonian. This means that the stable
higher derivative theory is characterized by two ingredients: the
bounded conserved quantity, and the Poisson bracket on its phase
space that brings the equations of motion to the Hamiltonian form
with a bounded from below Hamiltonian.
